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A Shape Dynamical Approach to Holographic Renormalization

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A Shape Dynamical Approach to Holographic Renormalization View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Open Access Repository Eur. Phys. J. C (2015) 75:3 DOI 10.1140/epjc/s10052-014-3238-z Regular Article - Theoretical Physics A shape dynamical approach to holographic renormalization Henrique Gomes1,a, Sean Gryb2,3,b, Tim Koslowski4,c, Flavio Mercati5,d, Lee Smolin5,e 1 University of California at Davis, One Shields Avenue, Davis, CA 95616, USA 2 Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 3 Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Nijmegen, The Netherlands 4 University of New Brunswick, Fredericton, NB E3B 5A3, Canada 5 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada Received: 7 October 2014 / Accepted: 18 December 2014 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract We provide a bottom-up argument to derive some 1.3 Main results .................... known results from holographic renormalization using the Notation ...................... classical bulk–bulk equivalence of General Relativity and 2 Mechanism ....................... Shape Dynamics, a theory with spatial conformal (Weyl) 2.1 The basic idea of gauge symmetry trading in invariance. The purpose of this paper is twofold: (1) to adver- General Relativity ................. tise the simple classical mechanism, trading off gauge sym- 2.2 Shape Dynamics description of gravity ...... metries, that underlies the bulk–bulk equivalence of General 2.3 Special CMC slices ................ Relativity and Shape Dynamics to readers interested in dual- 3 Asymptotically locally AdS and VPCT ........ ities of the type of AdS/conformal field theory (CFT); and 4 Holographic renormalization and Shape Dynamics .. (2) to highlight that this mechanism can be used to explain 4.1 Classical Shape Dynamics at large volume .... certain results of holographic renormalization, providing an 4.1.1 Volume expansion ............. alternative to the AdS/CFT conjecture for these cases. To 4.1.2 Hamilton–Jacobi equation ......... make contact with the usual semiclassical AdS/CFT corre- 4.2 Comparison with holographic renormalization spondence, we provide, in addition, a heuristic argument that results ....................... makes it plausible that the classical equivalence between 5 Remarks on Wilsonian renormalization ........ General Relativity and Shape Dynamics turns into a dual- 5.1 Generic recipe ................... ity between radial evolution in gravity and the renormaliza- 6 Conclusions ...................... tion group flow of a CFT. We believe that Shape Dynamics Appendix A: general mechanism for gauge symmetry provides a new perspective on gravity by giving conformal trading ......................... structure a primary role within the theory. It is hoped that this A.1. Symmetry trading and linking gauge theories .. work provides the first steps toward understanding what this A.2. Dictionary and observable equivalence ...... new perspective may be able to teach us about holographic A.3. Construction of linking gauge theories ...... dualities. Appendix B: scalar field (d ≤ 4) ............. References ......................... Contents 1 Introduction 1 Introduction ...................... 1.1 Heuristics ..................... A key insight of holography is that the behavior of a con- 1.2 Shape Dynamics ................. formal field theory (CFT) under renormalization is gov- erned by a diffeomorphism-invariant gravitational theory in a e-mail: [email protected] one higher spacetime dimension. In this correspondence, the b e-mail: [email protected] extra dimension plays the role of the renormalization scale. c e-mail: [email protected] This gives the renormalization group a geometric setting and d e-mail: [email protected] suggests deep connections between the dynamics of space- e e-mail: [email protected] time and the renormalization of quantum fields. 123 3 Page 2 of 15 Eur. Phys. J. C (2015) 75:3 The holographic formulation of the renormalization group which allows us to posit correspondences between them? was elucidated in a series of beautiful papers [1–9] and under- One well-known answer builds on global symmetries, in par- lies much of the very fertile applications of holography and ticular, the correspondence between the AdS group in d + 1 the AdS/CFT correspondence [10–12] to condensed matter dimensions and the global conformal group in d dimensions. and fluid systems. In this paper, we offer new insights into Here we would like to suggest that a deeper correspondence the connection between gravitation and renormalization by holds between gauge symmetries, in particular between the giving a new explanation for why that connection exists. We group of refoliations of the spacetime manifold in d + 1 give a novel derivation of a correspondence between a CFT dimensions and the group of Weyl transformations acting to defined on a d-dimensional manifold with metric Gab, locally rescale the metric on its boundary. This connection and a solution to a gravitational theory in d + 1 dimensions. between refoliations and Weyl transformations is captured In this correspondence, is to be interpreted as the asymp- by the mechanism of gauge symmetry trading which is at totic boundary of a Euclidean asymptotically AdS spacetime, the heart of Shape Dynamics. The main idea we want to M, the latter being a solution to a spacetime diffeomorphism develop in this paper is that it is the existence of gauge sym- invariant theory of a metric gμν such that = ∂M and Gab metry trading between refoliations of a d + 1 dimensional is the pullback of gμν onto .1 We give a plausibility argu- spacetime and Weyl transformations on fixed foliations of ment for the conjecture that this correspondence can be seen that spacetime that provide a deep and very general reason as the reason why gravitational evolution encodes the renor- why there exist correspondences between bulk gravitational malization group flow of a CFT, at least to leading order theories and boundary CFTs.2 in a semiclassical regime, near the conformal boundary of a The main work of this paper consists of showing how Euclidean locally asymptotically AdS spacetime. trading of spacetime refoliation invariance for spatial Weyl The new derivation we offer makes use of a recent refor- invariance can be used to explain a subset (which will mulation of General Relativity in which spatial conformal be explicitly specified later) of the results of holographic invariance plays a central role. This formulation, called renormalization. This trading of spacetime refoliation invari- Shape Dynamics, is the result of trading the many fingered ance for spatial Weyl invariance has some immediate con- time aspect of spacetime diffeomorphism invariance—which sequences for usual CFTs, since invariance of a field the- allows one to arbitrarily redefine what is meant by surfaces ory under Weyl transformations for a general metric implies of constant time—with local Weyl transformations on a fixed invariance under the global conformal group when that met- class of spatial surfaces. In a word, Shape Dynamics trades ric is (conformally) flat [13]. relativity of time for relativity of scale. Each is a gauge invari- We will see that it is very useful to consider separately ance that can be defined on the phase space of general relativ- constant scalings that affect the total spatial volume, and to ity depending on one free function. The purpose of this paper distinguish these from Weyl transformations that leave the is not only to re-derive some known results through a novel spatial volume fixed. As we will see, these volume preserv- route, but also to highlight a mechanism which is an alterna- ing conformal transformations (VPCT) play a special role in tive to the AdS/CFT conjecture, which can be used to explain Shape Dynamics. some results that are usually attributed to the AdS/CFT con- jecture. This mechanism is the trading of classical gauge symmetries, which we will explain in Sect. 2. The purpose 1.2 Shape Dynamics of this paper is thus to show that this mechanism is very gen- eral and that this general mechanism can be used to derive The main argument of this paper is based on number of obser- known results without using the AdS/CFT conjecture. An vations made during the development of the Shape Dynamics aspiration of the current work is that it lay down the founda- description of General Relativity, which was in part used in tions for exploring further interesting connections between [14]. These observations are: Shape Dynamics and holography. 1. There is a classical mechanism (gauge symmetry trad- 1.1 Heuristics ing), based on two partial gauge fixings of a linking gauge theory, which generates exact dualities between classical Before we posit a particular correspondence between a CFT gauge theories [15]. and a particular gravitational theory in one higher dimen- sion, we can ask a more general question: what properties do generic gravitational theories and generic CFT’s share 2 Note that it is already known that if a classical field theory is Weyl invariant for a general metric, hab the theory will be invariant under 1 We will highlight the differences and similarities between ours and global conformal transformations when that metric is taken to be the usual holographic renormalization program when they arise. flat [13]. 123 Eur. Phys. J. C (2015) 75:3 Page 3 of
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